2024/01/04 by Peruyero, H. Contreras, Suárez-Serrato, P. · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2401.02560
We show that the fundamental groups of smooth 4-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to 4 when aspherical. We also show that closed 3-manifold groups have asymptotic dimension at most 3. Our proof method yields that the asymptotic dimension of closed 3-dimensional Alexandrov spaces is at most 3. We thus obtain that the Novikov conjecture holds for closed 4-manifolds with such a geometric decomposition and closed 3-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain 0-surgered geometric 4-manifolds and the existence of zero in the spectrum of aspherical smooth 4-manifolds with a geometric decomposition.